It has been suggested that the ground-state magnetic phase of \(\mathrm {Mn_3 Ge}\)  is described by two in-plane vector orders, i.e., an antichiral (AC) order [Fig. REF (c)] and a weak ferromagnetic (FM) order [Fig. REF (d)] [1]}, [2]}. The AC order reduces the magnetic symmetry from hexagonal \(D_{6h}\)  to rhombic \(D_{2h}\)  and the FM order furthermore reduces \(D_{2h}\)  to monoclinic \(C_{2h}\)  [3]}. Assuming a uniform spin configuration respecting the translation and inversion symmetries of the \(\mathrm {Mn_3 Ge}\)  lattice, we consider the effective triple-spin JDC Hamiltonian describing the energy per unit cell [4]}, [2]}, [1]},
\(\mathcal {H} = &\, J \sum _{\langle jk \rangle } \mathbf {S}_j \cdot \mathbf {S}_k + D \sum _{\langle jk \rangle } \hat{\mathbf {z}} \cdot (\mathbf {S}_j \times \mathbf {S}_k) \nonumber \\&\, + \sum _{l=1}^{3}\sum _{m=-l}^{l} C_{2 l}^{2m} T_{2l}^{2m} - 2g\mu _\mathrm {B} \mathbf {H} \cdot \sum _{j} \mathbf {S}_j, \) 
